On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem

Eric Olivier (1)

1 I2M - Institut de Mathématiques de Marseille

Let 1 β G be the closed projection on the 2-torus of the (modified) Rademacher graph in base β. The smallest compact containing G and left invariant by the diagonal endomorphism $${(x,y)\mapsto(2x,\beta y)}$$ (mod 1) is denoted by K. For β a simple Parry number of PV-type, K is proved to be a sofic affine invariant set with a fractal geometry closed to the one of G. When β is the golden number, we prove the uniqueness of the measure with full Hausdorff dimension on K.

Domaines: Théorie des nombres [math.NT]

Eric Olivier. On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem. Monatshefte für Mathematik, 2012, 165 (3-4), pp.447--497. ⟨10.1007/s00605-011-0296-2⟩. ⟨hal-01387010⟩